Viscosity I

The most fundamental property of a fluid is (doh!) flow. Under application of a force, a fluid deforms continuously. That doesn’t mean that fluids don’t protest a bit, however. Pushing through a fluid usually produces a resistance – you need to do work to make it flow, and a flowing fluid produces forces on objects that it flows past. Of course fluids have mass, and changing the velocity of a mass requires a force, but I’m not talking about that kind of inertial force at the moment, but about the frictional force a fluid moving at constant velocity exerts on anything it flows past. That frictional quality is called viscosity.

We are familiar with very viscous fluids like honey, and much less viscous fluids, like water and air. Imagine stepping into a shallow but swift flowing stream. You feel a force as you insert your foot. That initial force can be explained purely on inertial grounds – you have forced the stream to change direction to flow around you. If the fluid were frictionless, or perfectly inviscid, that force would be perfectly balanced by an opposite force pushing on the other side of your foot as the stream came together behind it. Real streams don’t behave that way, of course, and you will continue to feel the force of the fluid even after your feet are planted and still.

So how does that friction work? Like other frictional forces, viscous friction (in its most familiar form) transfers momentum in directions transverse to the flow. There is more than one way this can happen. Some viscous fluids contain long chain molecules, or molecules that form temporary structures that extent transversely to the flow, or tend to stick to whatever they flow past. This causes the moving fluid to tug along on adjacent layers as it moves. Air isn’t like that, but it still is viscous, because molecules in the flow are moving randomly in directions transverse to the flow. Thus, if we imagine two layers of fluid moving past each other, molecules moving randomly from one layer to the next carry some of their momentum with them, thus transferring some of the momentum of that layer to the next.

The general concept of viscosity relates the forces on a fluid element (as described by the second rank stress tensor) to the deformation of that fluid element (as described by the second rank rate of deformation tensor) and hence is a fourth rank tensor. Lots of familiar fluids, including water and air, are Newtonian, and the viscosity simplifies to a single number.


One of the more fundamental notions for the mechanics of a viscous fluid is the Reynolds number, which is the ratio of the product of a characteristic length L times a characteristic Velocity divided by the kinematic viscosity (itself the ration of viscosity to density). The kinematic viscosity of air is about 2 x 10^(-5) m^2/s. Hence, for a car of length 5 m. moving at 20 m/s, the Reynolds number is 5x20/(2x10^-5 = 5,000,000. The Reynolds number is important mostly because it tells you the ratio of inertial forces governing fluid flow acting to the viscous forces.

One of the things this says is that movement through a fluid looks different depending on your size and speed. The smaller you are, and the slower you go, the more “viscous” the fluid looks, in the sense that the viscous forces become more important. Reynolds numbers for large jet aircraft are hundreds of millions, 15,000 or so for a hummingbird, perhaps only 100 for a fruit fly, and maybe 10 or less for the very smallest flying insects of all, some tiny parasitic wasps. Small fish swim at Re = 1 or so, and tiny bacteria at perhaps Re = 10^(-5). These littlest swimmers live in Aristotelian physics – the moment they stop swimming, they stop – no coasting allowed.

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